LMFDB - Embedded newform 6004.2.a.g.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$47.9421813736$
Analytic rank:	$1$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.604994 q^{3} +2.44978 q^{5} +2.35442 q^{7} -2.63398 q^{9} +O(q^{10})$	$q-0.604994 q^{3} +2.44978 q^{5} +2.35442 q^{7} -2.63398 q^{9} +0.468519 q^{11} +3.50927 q^{13} -1.48210 q^{15} -6.18064 q^{17} +1.00000 q^{19} -1.42441 q^{21} -7.62373 q^{23} +1.00144 q^{25} +3.40852 q^{27} +1.30072 q^{29} -1.64624 q^{31} -0.283451 q^{33} +5.76782 q^{35} -9.36695 q^{37} -2.12309 q^{39} -0.982363 q^{41} -10.0121 q^{43} -6.45269 q^{45} -2.43534 q^{47} -1.45671 q^{49} +3.73925 q^{51} +2.13707 q^{53} +1.14777 q^{55} -0.604994 q^{57} -14.8492 q^{59} -6.19170 q^{61} -6.20150 q^{63} +8.59696 q^{65} +9.26754 q^{67} +4.61231 q^{69} -2.60754 q^{71} +4.21490 q^{73} -0.605863 q^{75} +1.10309 q^{77} -1.00000 q^{79} +5.83981 q^{81} -2.00505 q^{83} -15.1412 q^{85} -0.786928 q^{87} +3.76515 q^{89} +8.26231 q^{91} +0.995968 q^{93} +2.44978 q^{95} -4.45794 q^{97} -1.23407 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.604994	−0.349293	−0.174647	−	0.984631i	$-0.555878\pi$
$3$	−0.604994	−0.349293	−0.174647	+	0.984631i	$0.555878\pi$
$4$	0	0
$5$	2.44978	1.09558	0.547788	−	0.836617i	$-0.315470\pi$
$5$	2.44978	1.09558	0.547788	+	0.836617i	$0.315470\pi$
$6$	0	0
$7$	2.35442	0.889887	0.444944	−	0.895559i	$-0.353224\pi$
$7$	2.35442	0.889887	0.444944	+	0.895559i	$0.353224\pi$
$8$	0	0
$9$	−2.63398	−0.877994
$10$	0	0
$11$	0.468519	0.141264	0.0706318	−	0.997502i	$-0.477498\pi$
$11$	0.468519	0.141264	0.0706318	+	0.997502i	$0.477498\pi$
$12$	0	0
$13$	3.50927	0.973298	0.486649	−	0.873598i	$-0.338219\pi$
$13$	3.50927	0.973298	0.486649	+	0.873598i	$0.338219\pi$
$14$	0	0
$15$	−1.48210	−0.382677
$16$	0	0
$17$	−6.18064	−1.49902	−0.749512	−	0.661990i	$-0.769712\pi$
$17$	−6.18064	−1.49902	−0.749512	+	0.661990i	$0.769712\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.42441	−0.310832
$22$	0	0
$23$	−7.62373	−1.58966	−0.794829	−	0.606833i	$-0.792440\pi$
$23$	−7.62373	−1.58966	−0.794829	+	0.606833i	$0.792440\pi$
$24$	0	0
$25$	1.00144	0.200287
$26$	0	0
$27$	3.40852	0.655971
$28$	0	0
$29$	1.30072	0.241538	0.120769	−	0.992681i	$-0.461464\pi$
$29$	1.30072	0.241538	0.120769	+	0.992681i	$0.461464\pi$
$30$	0	0
$31$	−1.64624	−0.295674	−0.147837	−	0.989012i	$-0.547231\pi$
$31$	−1.64624	−0.295674	−0.147837	+	0.989012i	$0.547231\pi$
$32$	0	0
$33$	−0.283451	−0.0493424
$34$	0	0
$35$	5.76782	0.974939
$36$	0	0
$37$	−9.36695	−1.53992	−0.769959	−	0.638093i	$-0.779723\pi$
$37$	−9.36695	−1.53992	−0.769959	+	0.638093i	$0.779723\pi$
$38$	0	0
$39$	−2.12309	−0.339966
$40$	0	0
$41$	−0.982363	−0.153419	−0.0767096	−	0.997053i	$-0.524441\pi$
$41$	−0.982363	−0.153419	−0.0767096	+	0.997053i	$0.524441\pi$
$42$	0	0
$43$	−10.0121	−1.52683	−0.763416	−	0.645907i	$-0.776479\pi$
$43$	−10.0121	−1.52683	−0.763416	+	0.645907i	$0.776479\pi$
$44$	0	0
$45$	−6.45269	−0.961910
$46$	0	0
$47$	−2.43534	−0.355231	−0.177615	−	0.984100i	$-0.556838\pi$
$47$	−2.43534	−0.355231	−0.177615	+	0.984100i	$0.556838\pi$
$48$	0	0
$49$	−1.45671	−0.208101
$50$	0	0
$51$	3.73925	0.523599
$52$	0	0
$53$	2.13707	0.293549	0.146774	−	0.989170i	$-0.453111\pi$
$53$	2.13707	0.293549	0.146774	+	0.989170i	$0.453111\pi$
$54$	0	0
$55$	1.14777	0.154765
$56$	0	0
$57$	−0.604994	−0.0801334
$58$	0	0
$59$	−14.8492	−1.93320	−0.966602	−	0.256283i	$-0.917502\pi$
$59$	−14.8492	−1.93320	−0.966602	+	0.256283i	$0.917502\pi$
$60$	0	0
$61$	−6.19170	−0.792766	−0.396383	−	0.918085i	$-0.629735\pi$
$61$	−6.19170	−0.792766	−0.396383	+	0.918085i	$0.629735\pi$
$62$	0	0
$63$	−6.20150	−0.781316
$64$	0	0
$65$	8.59696	1.06632
$66$	0	0
$67$	9.26754	1.13221	0.566105	−	0.824333i	$-0.308450\pi$
$67$	9.26754	1.13221	0.566105	+	0.824333i	$0.308450\pi$
$68$	0	0
$69$	4.61231	0.555257
$70$	0	0
$71$	−2.60754	−0.309458	−0.154729	−	0.987957i	$-0.549450\pi$
$71$	−2.60754	−0.309458	−0.154729	+	0.987957i	$0.549450\pi$
$72$	0	0
$73$	4.21490	0.493316	0.246658	−	0.969103i	$-0.420668\pi$
$73$	4.21490	0.493316	0.246658	+	0.969103i	$0.420668\pi$
$74$	0	0
$75$	−0.605863	−0.0699590
$76$	0	0
$77$	1.10309	0.125709
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	5.83981	0.648868
$82$	0	0
$83$	−2.00505	−0.220083	−0.110041	−	0.993927i	$-0.535098\pi$
$83$	−2.00505	−0.220083	−0.110041	+	0.993927i	$0.535098\pi$
$84$	0	0
$85$	−15.1412	−1.64230
$86$	0	0
$87$	−0.786928	−0.0843675
$88$	0	0
$89$	3.76515	0.399105	0.199553	−	0.979887i	$-0.436051\pi$
$89$	3.76515	0.399105	0.199553	+	0.979887i	$0.436051\pi$
$90$	0	0
$91$	8.26231	0.866125
$92$	0	0
$93$	0.995968	0.103277
$94$	0	0
$95$	2.44978	0.251342
$96$	0	0
$97$	−4.45794	−0.452635	−0.226318	−	0.974054i	$-0.572669\pi$
$97$	−4.45794	−0.452635	−0.226318	+	0.974054i	$0.572669\pi$
$98$	0	0
$99$	−1.23407	−0.124029
$100$	0	0
$101$	−4.81751	−0.479360	−0.239680	−	0.970852i	$-0.577043\pi$
$101$	−4.81751	−0.479360	−0.239680	+	0.970852i	$0.577043\pi$
$102$	0	0
$103$	13.7916	1.35892	0.679461	−	0.733712i	$-0.262214\pi$
$103$	13.7916	1.35892	0.679461	+	0.733712i	$0.262214\pi$
$104$	0	0
$105$	−3.48949	−0.340540
$106$	0	0
$107$	−14.6905	−1.42018	−0.710090	−	0.704111i	$-0.751346\pi$
$107$	−14.6905	−1.42018	−0.710090	+	0.704111i	$0.751346\pi$
$108$	0	0
$109$	−5.54311	−0.530934	−0.265467	−	0.964120i	$-0.585526\pi$
$109$	−5.54311	−0.530934	−0.265467	+	0.964120i	$0.585526\pi$
$110$	0	0
$111$	5.66695	0.537883
$112$	0	0
$113$	−8.12912	−0.764723	−0.382362	−	0.924013i	$-0.624889\pi$
$113$	−8.12912	−0.764723	−0.382362	+	0.924013i	$0.624889\pi$
$114$	0	0
$115$	−18.6765	−1.74159
$116$	0	0
$117$	−9.24337	−0.854550
$118$	0	0
$119$	−14.5518	−1.33396
$120$	0	0
$121$	−10.7805	−0.980045
$122$	0	0
$123$	0.594323	0.0535883
$124$	0	0
$125$	−9.79561	−0.876146
$126$	0	0
$127$	2.64614	0.234807	0.117404	−	0.993084i	$-0.462543\pi$
$127$	2.64614	0.234807	0.117404	+	0.993084i	$0.462543\pi$
$128$	0	0
$129$	6.05726	0.533312
$130$	0	0
$131$	−2.88210	−0.251810	−0.125905	−	0.992042i	$-0.540183\pi$
$131$	−2.88210	−0.251810	−0.125905	+	0.992042i	$0.540183\pi$
$132$	0	0
$133$	2.35442	0.204154
$134$	0	0
$135$	8.35014	0.718666
$136$	0	0
$137$	12.2618	1.04760	0.523799	−	0.851842i	$-0.324514\pi$
$137$	12.2618	1.04760	0.523799	+	0.851842i	$0.324514\pi$
$138$	0	0
$139$	11.9899	1.01697	0.508483	−	0.861072i	$-0.330206\pi$
$139$	11.9899	1.01697	0.508483	+	0.861072i	$0.330206\pi$
$140$	0	0
$141$	1.47336	0.124080
$142$	0	0
$143$	1.64416	0.137492
$144$	0	0
$145$	3.18648	0.264623
$146$	0	0
$147$	0.881298	0.0726882
$148$	0	0
$149$	12.4904	1.02325	0.511626	−	0.859208i	$-0.329043\pi$
$149$	12.4904	1.02325	0.511626	+	0.859208i	$0.329043\pi$
$150$	0	0
$151$	−3.55007	−0.288900	−0.144450	−	0.989512i	$-0.546141\pi$
$151$	−3.55007	−0.288900	−0.144450	+	0.989512i	$0.546141\pi$
$152$	0	0
$153$	16.2797	1.31614
$154$	0	0
$155$	−4.03294	−0.323934
$156$	0	0
$157$	2.25373	0.179867	0.0899336	−	0.995948i	$-0.471335\pi$
$157$	2.25373	0.179867	0.0899336	+	0.995948i	$0.471335\pi$
$158$	0	0
$159$	−1.29291	−0.102535
$160$	0	0
$161$	−17.9495	−1.41462
$162$	0	0
$163$	17.1560	1.34376	0.671881	−	0.740659i	$-0.265487\pi$
$163$	17.1560	1.34376	0.671881	+	0.740659i	$0.265487\pi$
$164$	0	0
$165$	−0.694393	−0.0540584
$166$	0	0
$167$	4.87658	0.377361	0.188681	−	0.982038i	$-0.439579\pi$
$167$	4.87658	0.377361	0.188681	+	0.982038i	$0.439579\pi$
$168$	0	0
$169$	−0.684990	−0.0526915
$170$	0	0
$171$	−2.63398	−0.201426
$172$	0	0
$173$	0.524244	0.0398575	0.0199288	−	0.999801i	$-0.493656\pi$
$173$	0.524244	0.0398575	0.0199288	+	0.999801i	$0.493656\pi$
$174$	0	0
$175$	2.35780	0.178233
$176$	0	0
$177$	8.98368	0.675255
$178$	0	0
$179$	6.19268	0.462863	0.231431	−	0.972851i	$-0.425659\pi$
$179$	6.19268	0.462863	0.231431	+	0.972851i	$0.425659\pi$
$180$	0	0
$181$	3.98028	0.295852	0.147926	−	0.988998i	$-0.452740\pi$
$181$	3.98028	0.295852	0.147926	+	0.988998i	$0.452740\pi$
$182$	0	0
$183$	3.74594	0.276908
$184$	0	0
$185$	−22.9470	−1.68710
$186$	0	0
$187$	−2.89574	−0.211758
$188$	0	0
$189$	8.02510	0.583740
$190$	0	0
$191$	−10.4934	−0.759279	−0.379639	−	0.925135i	$-0.623952\pi$
$191$	−10.4934	−0.759279	−0.379639	+	0.925135i	$0.623952\pi$
$192$	0	0
$193$	10.8655	0.782115	0.391057	−	0.920366i	$-0.372109\pi$
$193$	10.8655	0.782115	0.391057	+	0.920366i	$0.372109\pi$
$194$	0	0
$195$	−5.20111	−0.372459
$196$	0	0
$197$	−10.6489	−0.758702	−0.379351	−	0.925253i	$-0.623853\pi$
$197$	−10.6489	−0.758702	−0.379351	+	0.925253i	$0.623853\pi$
$198$	0	0
$199$	22.1387	1.56937	0.784684	−	0.619895i	$-0.212825\pi$
$199$	22.1387	1.56937	0.784684	+	0.619895i	$0.212825\pi$
$200$	0	0
$201$	−5.60680	−0.395473
$202$	0	0
$203$	3.06244	0.214941
$204$	0	0
$205$	−2.40657	−0.168082
$206$	0	0
$207$	20.0808	1.39571
$208$	0	0
$209$	0.468519	0.0324081
$210$	0	0
$211$	4.53308	0.312070	0.156035	−	0.987752i	$-0.450129\pi$
$211$	4.53308	0.312070	0.156035	+	0.987752i	$0.450129\pi$
$212$	0	0
$213$	1.57754	0.108092
$214$	0	0
$215$	−24.5275	−1.67276
$216$	0	0
$217$	−3.87595	−0.263117
$218$	0	0
$219$	−2.54999	−0.172312
$220$	0	0
$221$	−21.6896	−1.45900
$222$	0	0
$223$	22.3393	1.49595	0.747976	−	0.663726i	$-0.231026\pi$
$223$	22.3393	1.49595	0.747976	+	0.663726i	$0.231026\pi$
$224$	0	0
$225$	−2.63777	−0.175851
$226$	0	0
$227$	1.81604	0.120535	0.0602675	−	0.998182i	$-0.480805\pi$
$227$	1.81604	0.120535	0.0602675	+	0.998182i	$0.480805\pi$
$228$	0	0
$229$	−17.0168	−1.12450	−0.562250	−	0.826967i	$-0.690064\pi$
$229$	−17.0168	−1.12450	−0.562250	+	0.826967i	$0.690064\pi$
$230$	0	0
$231$	−0.667362	−0.0439092
$232$	0	0
$233$	−3.65045	−0.239149	−0.119575	−	0.992825i	$-0.538153\pi$
$233$	−3.65045	−0.239149	−0.119575	+	0.992825i	$0.538153\pi$
$234$	0	0
$235$	−5.96605	−0.389182
$236$	0	0
$237$	0.604994	0.0392986
$238$	0	0
$239$	19.3988	1.25481	0.627403	−	0.778694i	$-0.284118\pi$
$239$	19.3988	1.25481	0.627403	+	0.778694i	$0.284118\pi$
$240$	0	0
$241$	−9.13327	−0.588326	−0.294163	−	0.955755i	$-0.595041\pi$
$241$	−9.13327	−0.588326	−0.294163	+	0.955755i	$0.595041\pi$
$242$	0	0
$243$	−13.7586	−0.882616
$244$	0	0
$245$	−3.56861	−0.227990
$246$	0	0
$247$	3.50927	0.223290
$248$	0	0
$249$	1.21304	0.0768734
$250$	0	0
$251$	1.08945	0.0687652	0.0343826	−	0.999409i	$-0.489054\pi$
$251$	1.08945	0.0687652	0.0343826	+	0.999409i	$0.489054\pi$
$252$	0	0
$253$	−3.57186	−0.224561
$254$	0	0
$255$	9.16034	0.573643
$256$	0	0
$257$	12.2335	0.763105	0.381553	−	0.924347i	$-0.375389\pi$
$257$	12.2335	0.763105	0.381553	+	0.924347i	$0.375389\pi$
$258$	0	0
$259$	−22.0537	−1.37035
$260$	0	0
$261$	−3.42608	−0.212069
$262$	0	0
$263$	19.6933	1.21434	0.607171	−	0.794571i	$-0.292304\pi$
$263$	19.6933	1.21434	0.607171	+	0.794571i	$0.292304\pi$
$264$	0	0
$265$	5.23535	0.321605
$266$	0	0
$267$	−2.27789	−0.139405
$268$	0	0
$269$	1.74189	0.106205	0.0531025	−	0.998589i	$-0.483089\pi$
$269$	1.74189	0.106205	0.0531025	+	0.998589i	$0.483089\pi$
$270$	0	0
$271$	25.0149	1.51954	0.759772	−	0.650190i	$-0.225311\pi$
$271$	25.0149	1.51954	0.759772	+	0.650190i	$0.225311\pi$
$272$	0	0
$273$	−4.99864	−0.302532
$274$	0	0
$275$	0.469192	0.0282933
$276$	0	0
$277$	−5.35877	−0.321977	−0.160989	−	0.986956i	$-0.551468\pi$
$277$	−5.35877	−0.321977	−0.160989	+	0.986956i	$0.551468\pi$
$278$	0	0
$279$	4.33618	0.259600
$280$	0	0
$281$	−8.35419	−0.498369	−0.249185	−	0.968456i	$-0.580163\pi$
$281$	−8.35419	−0.498369	−0.249185	+	0.968456i	$0.580163\pi$
$282$	0	0
$283$	−7.65083	−0.454795	−0.227397	−	0.973802i	$-0.573022\pi$
$283$	−7.65083	−0.454795	−0.227397	+	0.973802i	$0.573022\pi$
$284$	0	0
$285$	−1.48210	−0.0877922
$286$	0	0
$287$	−2.31289	−0.136526
$288$	0	0
$289$	21.2003	1.24708
$290$	0	0
$291$	2.69703	0.158102
$292$	0	0
$293$	−0.0471832	−0.00275647	−0.00137824	−	0.999999i	$-0.500439\pi$
$293$	−0.0471832	−0.00275647	−0.00137824	+	0.999999i	$0.500439\pi$
$294$	0	0
$295$	−36.3774	−2.11797
$296$	0	0
$297$	1.59696	0.0926648
$298$	0	0
$299$	−26.7538	−1.54721
$300$	0	0
$301$	−23.5727	−1.35871
$302$	0	0
$303$	2.91456	0.167437
$304$	0	0
$305$	−15.1683	−0.868536
$306$	0	0
$307$	−24.7943	−1.41509	−0.707543	−	0.706670i	$-0.750196\pi$
$307$	−24.7943	−1.41509	−0.707543	+	0.706670i	$0.750196\pi$
$308$	0	0
$309$	−8.34380	−0.474662
$310$	0	0
$311$	−21.4159	−1.21439	−0.607193	−	0.794554i	$-0.707705\pi$
$311$	−21.4159	−1.21439	−0.607193	+	0.794554i	$0.707705\pi$
$312$	0	0
$313$	−16.1755	−0.914295	−0.457147	−	0.889391i	$-0.651129\pi$
$313$	−16.1755	−0.914295	−0.457147	+	0.889391i	$0.651129\pi$
$314$	0	0
$315$	−15.1923	−0.855991
$316$	0	0
$317$	−20.2817	−1.13913	−0.569567	−	0.821945i	$-0.692889\pi$
$317$	−20.2817	−1.13913	−0.569567	+	0.821945i	$0.692889\pi$
$318$	0	0
$319$	0.609412	0.0341205
$320$	0	0
$321$	8.88764	0.496059
$322$	0	0
$323$	−6.18064	−0.343900
$324$	0	0
$325$	3.51432	0.194939
$326$	0	0
$327$	3.35355	0.185452
$328$	0	0
$329$	−5.73381	−0.316115
$330$	0	0
$331$	9.99691	0.549480	0.274740	−	0.961519i	$-0.411408\pi$
$331$	9.99691	0.549480	0.274740	+	0.961519i	$0.411408\pi$
$332$	0	0
$333$	24.6724	1.35204
$334$	0	0
$335$	22.7035	1.24042
$336$	0	0
$337$	2.14801	0.117009	0.0585047	−	0.998287i	$-0.481367\pi$
$337$	2.14801	0.117009	0.0585047	+	0.998287i	$0.481367\pi$
$338$	0	0
$339$	4.91807	0.267113
$340$	0	0
$341$	−0.771296	−0.0417680
$342$	0	0
$343$	−19.9106	−1.07507
$344$	0	0
$345$	11.2992	0.608326
$346$	0	0
$347$	−3.67034	−0.197034	−0.0985170	−	0.995135i	$-0.531410\pi$
$347$	−3.67034	−0.197034	−0.0985170	+	0.995135i	$0.531410\pi$
$348$	0	0
$349$	18.2766	0.978322	0.489161	−	0.872193i	$-0.337303\pi$
$349$	18.2766	0.978322	0.489161	+	0.872193i	$0.337303\pi$
$350$	0	0
$351$	11.9614	0.638455
$352$	0	0
$353$	−33.9309	−1.80596	−0.902980	−	0.429683i	$-0.858625\pi$
$353$	−33.9309	−1.80596	−0.902980	+	0.429683i	$0.858625\pi$
$354$	0	0
$355$	−6.38790	−0.339035
$356$	0	0
$357$	8.80376	0.465944
$358$	0	0
$359$	−7.97748	−0.421035	−0.210518	−	0.977590i	$-0.567515\pi$
$359$	−7.97748	−0.421035	−0.210518	+	0.977590i	$0.567515\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	6.52213	0.342323
$364$	0	0
$365$	10.3256	0.540466
$366$	0	0
$367$	17.4424	0.910488	0.455244	−	0.890367i	$-0.349552\pi$
$367$	17.4424	0.910488	0.455244	+	0.890367i	$0.349552\pi$
$368$	0	0
$369$	2.58753	0.134701
$370$	0	0
$371$	5.03155	0.261225
$372$	0	0
$373$	−17.3519	−0.898449	−0.449224	−	0.893419i	$-0.648300\pi$
$373$	−17.3519	−0.898449	−0.449224	+	0.893419i	$0.648300\pi$
$374$	0	0
$375$	5.92628	0.306032
$376$	0	0
$377$	4.56459	0.235088
$378$	0	0
$379$	−16.4005	−0.842440	−0.421220	−	0.906959i	$-0.638398\pi$
$379$	−16.4005	−0.842440	−0.421220	+	0.906959i	$0.638398\pi$
$380$	0	0
$381$	−1.60090	−0.0820166
$382$	0	0
$383$	−25.2958	−1.29256	−0.646278	−	0.763102i	$-0.723675\pi$
$383$	−25.2958	−1.29256	−0.646278	+	0.763102i	$0.723675\pi$
$384$	0	0
$385$	2.70233	0.137723
$386$	0	0
$387$	26.3717	1.34055
$388$	0	0
$389$	0.142242	0.00721196	0.00360598	−	0.999993i	$-0.498852\pi$
$389$	0.142242	0.00721196	0.00360598	+	0.999993i	$0.498852\pi$
$390$	0	0
$391$	47.1195	2.38294
$392$	0	0
$393$	1.74365	0.0879555
$394$	0	0
$395$	−2.44978	−0.123262
$396$	0	0
$397$	16.7945	0.842893	0.421446	−	0.906853i	$-0.361523\pi$
$397$	16.7945	0.842893	0.421446	+	0.906853i	$0.361523\pi$
$398$	0	0
$399$	−1.42441	−0.0713096
$400$	0	0
$401$	16.1890	0.808438	0.404219	−	0.914662i	$-0.367543\pi$
$401$	16.1890	0.808438	0.404219	+	0.914662i	$0.367543\pi$
$402$	0	0
$403$	−5.77713	−0.287779
$404$	0	0
$405$	14.3063	0.710884
$406$	0	0
$407$	−4.38859	−0.217534
$408$	0	0
$409$	−21.0235	−1.03955	−0.519773	−	0.854304i	$-0.673984\pi$
$409$	−21.0235	−1.03955	−0.519773	+	0.854304i	$0.673984\pi$
$410$	0	0
$411$	−7.41833	−0.365919
$412$	0	0
$413$	−34.9613	−1.72033
$414$	0	0
$415$	−4.91194	−0.241117
$416$	0	0
$417$	−7.25379	−0.355220
$418$	0	0
$419$	−27.5659	−1.34668	−0.673342	−	0.739331i	$-0.735142\pi$
$419$	−27.5659	−1.34668	−0.673342	+	0.739331i	$0.735142\pi$
$420$	0	0
$421$	−7.54751	−0.367843	−0.183922	−	0.982941i	$-0.558879\pi$
$421$	−7.54751	−0.367843	−0.183922	+	0.982941i	$0.558879\pi$
$422$	0	0
$423$	6.41464	0.311890
$424$	0	0
$425$	−6.18952	−0.300236
$426$	0	0
$427$	−14.5779	−0.705473
$428$	0	0
$429$	−0.994707	−0.0480249
$430$	0	0
$431$	20.8660	1.00508	0.502540	−	0.864554i	$-0.332399\pi$
$431$	20.8660	1.00508	0.502540	+	0.864554i	$0.332399\pi$
$432$	0	0
$433$	−19.0135	−0.913731	−0.456865	−	0.889536i	$-0.651028\pi$
$433$	−19.0135	−0.913731	−0.456865	+	0.889536i	$0.651028\pi$
$434$	0	0
$435$	−1.92780	−0.0924310
$436$	0	0
$437$	−7.62373	−0.364693
$438$	0	0
$439$	19.8903	0.949312	0.474656	−	0.880171i	$-0.342572\pi$
$439$	19.8903	0.949312	0.474656	+	0.880171i	$0.342572\pi$
$440$	0	0
$441$	3.83694	0.182711
$442$	0	0
$443$	24.6162	1.16955	0.584776	−	0.811195i	$-0.301182\pi$
$443$	24.6162	1.16955	0.584776	+	0.811195i	$0.301182\pi$
$444$	0	0
$445$	9.22381	0.437250
$446$	0	0
$447$	−7.55660	−0.357415
$448$	0	0
$449$	−3.98532	−0.188079	−0.0940394	−	0.995568i	$-0.529978\pi$
$449$	−3.98532	−0.188079	−0.0940394	+	0.995568i	$0.529978\pi$
$450$	0	0
$451$	−0.460255	−0.0216726
$452$	0	0
$453$	2.14777	0.100911
$454$	0	0
$455$	20.2409	0.948906
$456$	0	0
$457$	−6.19826	−0.289942	−0.144971	−	0.989436i	$-0.546309\pi$
$457$	−6.19826	−0.289942	−0.144971	+	0.989436i	$0.546309\pi$
$458$	0	0
$459$	−21.0669	−0.983316
$460$	0	0
$461$	−10.1185	−0.471267	−0.235634	−	0.971842i	$-0.575716\pi$
$461$	−10.1185	−0.471267	−0.235634	+	0.971842i	$0.575716\pi$
$462$	0	0
$463$	27.4102	1.27386	0.636931	−	0.770921i	$-0.280204\pi$
$463$	27.4102	1.27386	0.636931	+	0.770921i	$0.280204\pi$
$464$	0	0
$465$	2.43990	0.113148
$466$	0	0
$467$	−7.55393	−0.349554	−0.174777	−	0.984608i	$-0.555921\pi$
$467$	−7.55393	−0.349554	−0.174777	+	0.984608i	$0.555921\pi$
$468$	0	0
$469$	21.8197	1.00754
$470$	0	0
$471$	−1.36349	−0.0628264
$472$	0	0
$473$	−4.69086	−0.215686
$474$	0	0
$475$	1.00144	0.0459491
$476$	0	0
$477$	−5.62899	−0.257734
$478$	0	0
$479$	19.6830	0.899340	0.449670	−	0.893195i	$-0.351542\pi$
$479$	19.6830	0.899340	0.449670	+	0.893195i	$0.351542\pi$
$480$	0	0
$481$	−32.8712	−1.49880
$482$	0	0
$483$	10.8593	0.494116
$484$	0	0
$485$	−10.9210	−0.495897
$486$	0	0
$487$	−22.1199	−1.00235	−0.501175	−	0.865346i	$-0.667099\pi$
$487$	−22.1199	−1.00235	−0.501175	+	0.865346i	$0.667099\pi$
$488$	0	0
$489$	−10.3793	−0.469367
$490$	0	0
$491$	8.54047	0.385426	0.192713	−	0.981255i	$-0.438271\pi$
$491$	8.54047	0.385426	0.192713	+	0.981255i	$0.438271\pi$
$492$	0	0
$493$	−8.03928	−0.362071
$494$	0	0
$495$	−3.02320	−0.135883
$496$	0	0
$497$	−6.13924	−0.275383
$498$	0	0
$499$	−31.9166	−1.42878	−0.714391	−	0.699746i	$-0.753296\pi$
$499$	−31.9166	−1.42878	−0.714391	+	0.699746i	$0.753296\pi$
$500$	0	0
$501$	−2.95030	−0.131810
$502$	0	0
$503$	1.31642	0.0586964	0.0293482	−	0.999569i	$-0.490657\pi$
$503$	1.31642	0.0586964	0.0293482	+	0.999569i	$0.490657\pi$
$504$	0	0
$505$	−11.8018	−0.525175
$506$	0	0
$507$	0.414414	0.0184048
$508$	0	0
$509$	−1.23657	−0.0548101	−0.0274050	−	0.999624i	$-0.508724\pi$
$509$	−1.23657	−0.0548101	−0.0274050	+	0.999624i	$0.508724\pi$
$510$	0	0
$511$	9.92364	0.438996
$512$	0	0
$513$	3.40852	0.150490
$514$	0	0
$515$	33.7863	1.48880
$516$	0	0
$517$	−1.14100	−0.0501812
$518$	0	0
$519$	−0.317164	−0.0139220
$520$	0	0
$521$	−19.1777	−0.840192	−0.420096	−	0.907480i	$-0.638004\pi$
$521$	−19.1777	−0.840192	−0.420096	+	0.907480i	$0.638004\pi$
$522$	0	0
$523$	16.4298	0.718427	0.359213	−	0.933255i	$-0.383045\pi$
$523$	16.4298	0.718427	0.359213	+	0.933255i	$0.383045\pi$
$524$	0	0
$525$	−1.42646	−0.0622556
$526$	0	0
$527$	10.1748	0.443223
$528$	0	0
$529$	35.1213	1.52701
$530$	0	0
$531$	39.1126	1.69734
$532$	0	0
$533$	−3.44738	−0.149323
$534$	0	0
$535$	−35.9884	−1.55592
$536$	0	0
$537$	−3.74653	−0.161675
$538$	0	0
$539$	−0.682494	−0.0293971
$540$	0	0
$541$	2.33319	0.100311	0.0501557	−	0.998741i	$-0.484028\pi$
$541$	2.33319	0.100311	0.0501557	+	0.998741i	$0.484028\pi$
$542$	0	0
$543$	−2.40804	−0.103339
$544$	0	0
$545$	−13.5794	−0.581679
$546$	0	0
$547$	22.6740	0.969470	0.484735	−	0.874661i	$-0.338916\pi$
$547$	22.6740	0.969470	0.484735	+	0.874661i	$0.338916\pi$
$548$	0	0
$549$	16.3088	0.696044
$550$	0	0
$551$	1.30072	0.0554126
$552$	0	0
$553$	−2.35442	−0.100120
$554$	0	0
$555$	13.8828	0.589292
$556$	0	0
$557$	6.16892	0.261386	0.130693	−	0.991423i	$-0.458280\pi$
$557$	6.16892	0.261386	0.130693	+	0.991423i	$0.458280\pi$
$558$	0	0
$559$	−35.1352	−1.48606
$560$	0	0
$561$	1.75191	0.0739655
$562$	0	0
$563$	18.8505	0.794453	0.397227	−	0.917721i	$-0.369973\pi$
$563$	18.8505	0.794453	0.397227	+	0.917721i	$0.369973\pi$
$564$	0	0
$565$	−19.9146	−0.837813
$566$	0	0
$567$	13.7494	0.577419
$568$	0	0
$569$	−35.7486	−1.49866	−0.749331	−	0.662196i	$-0.769625\pi$
$569$	−35.7486	−1.49866	−0.749331	+	0.662196i	$0.769625\pi$
$570$	0	0
$571$	−24.3618	−1.01951	−0.509755	−	0.860320i	$-0.670264\pi$
$571$	−24.3618	−1.01951	−0.509755	+	0.860320i	$0.670264\pi$
$572$	0	0
$573$	6.34847	0.265211
$574$	0	0
$575$	−7.63469	−0.318388
$576$	0	0
$577$	−35.5817	−1.48129	−0.740643	−	0.671899i	$-0.765479\pi$
$577$	−35.5817	−1.48129	−0.740643	+	0.671899i	$0.765479\pi$
$578$	0	0
$579$	−6.57355	−0.273187
$580$	0	0
$581$	−4.72073	−0.195849
$582$	0	0
$583$	1.00125	0.0414677
$584$	0	0
$585$	−22.6442	−0.936224
$586$	0	0
$587$	4.65457	0.192115	0.0960574	−	0.995376i	$-0.469377\pi$
$587$	4.65457	0.192115	0.0960574	+	0.995376i	$0.469377\pi$
$588$	0	0
$589$	−1.64624	−0.0678323
$590$	0	0
$591$	6.44251	0.265010
$592$	0	0
$593$	−38.0844	−1.56394	−0.781970	−	0.623316i	$-0.785785\pi$
$593$	−38.0844	−1.56394	−0.781970	+	0.623316i	$0.785785\pi$
$594$	0	0
$595$	−35.6488	−1.46146
$596$	0	0
$597$	−13.3938	−0.548170
$598$	0	0
$599$	21.2478	0.868163	0.434082	−	0.900874i	$-0.357073\pi$
$599$	21.2478	0.868163	0.434082	+	0.900874i	$0.357073\pi$
$600$	0	0
$601$	−27.2110	−1.10996	−0.554980	−	0.831863i	$-0.687274\pi$
$601$	−27.2110	−1.10996	−0.554980	+	0.831863i	$0.687274\pi$
$602$	0	0
$603$	−24.4105	−0.994074
$604$	0	0
$605$	−26.4099	−1.07371
$606$	0	0
$607$	−20.8955	−0.848122	−0.424061	−	0.905634i	$-0.639396\pi$
$607$	−20.8955	−0.848122	−0.424061	+	0.905634i	$0.639396\pi$
$608$	0	0
$609$	−1.85276	−0.0750776
$610$	0	0
$611$	−8.54627	−0.345745
$612$	0	0
$613$	−10.2749	−0.414999	−0.207499	−	0.978235i	$-0.566532\pi$
$613$	−10.2749	−0.414999	−0.207499	+	0.978235i	$0.566532\pi$
$614$	0	0
$615$	1.45596	0.0587101
$616$	0	0
$617$	−2.43149	−0.0978881	−0.0489441	−	0.998802i	$-0.515586\pi$
$617$	−2.43149	−0.0978881	−0.0489441	+	0.998802i	$0.515586\pi$
$618$	0	0
$619$	−22.9624	−0.922937	−0.461468	−	0.887157i	$-0.652677\pi$
$619$	−22.9624	−0.922937	−0.461468	+	0.887157i	$0.652677\pi$
$620$	0	0
$621$	−25.9857	−1.04277
$622$	0	0
$623$	8.86475	0.355159
$624$	0	0
$625$	−29.0043	−1.16017
$626$	0	0
$627$	−0.283451	−0.0113199
$628$	0	0
$629$	57.8938	2.30838
$630$	0	0
$631$	−3.41002	−0.135751	−0.0678754	−	0.997694i	$-0.521622\pi$
$631$	−3.41002	−0.135751	−0.0678754	+	0.997694i	$0.521622\pi$
$632$	0	0
$633$	−2.74248	−0.109004
$634$	0	0
$635$	6.48248	0.257249
$636$	0	0
$637$	−5.11198	−0.202544
$638$	0	0
$639$	6.86821	0.271702
$640$	0	0
$641$	−18.8334	−0.743876	−0.371938	−	0.928258i	$-0.621307\pi$
$641$	−18.8334	−0.743876	−0.371938	+	0.928258i	$0.621307\pi$
$642$	0	0
$643$	11.4571	0.451824	0.225912	−	0.974148i	$-0.427464\pi$
$643$	11.4571	0.451824	0.225912	+	0.974148i	$0.427464\pi$
$644$	0	0
$645$	14.8390	0.584284
$646$	0	0
$647$	−18.3098	−0.719831	−0.359916	−	0.932985i	$-0.617195\pi$
$647$	−18.3098	−0.719831	−0.359916	+	0.932985i	$0.617195\pi$
$648$	0	0
$649$	−6.95713	−0.273091
$650$	0	0
$651$	2.34493	0.0919049
$652$	0	0
$653$	50.0215	1.95749	0.978747	−	0.205072i	$-0.0657428\pi$
$653$	50.0215	1.95749	0.978747	+	0.205072i	$0.0657428\pi$
$654$	0	0
$655$	−7.06051	−0.275877
$656$	0	0
$657$	−11.1020	−0.433129
$658$	0	0
$659$	46.6649	1.81780	0.908902	−	0.417009i	$-0.136922\pi$
$659$	46.6649	1.81780	0.908902	+	0.417009i	$0.136922\pi$
$660$	0	0
$661$	−44.0677	−1.71404	−0.857018	−	0.515287i	$-0.827685\pi$
$661$	−44.0677	−1.71404	−0.857018	+	0.515287i	$0.827685\pi$
$662$	0	0
$663$	13.1220	0.509618
$664$	0	0
$665$	5.76782	0.223666
$666$	0	0
$667$	−9.91635	−0.383963
$668$	0	0
$669$	−13.5151	−0.522526
$670$	0	0
$671$	−2.90093	−0.111989
$672$	0	0
$673$	−3.44752	−0.132892	−0.0664461	−	0.997790i	$-0.521166\pi$
$673$	−3.44752	−0.132892	−0.0664461	+	0.997790i	$0.521166\pi$
$674$	0	0
$675$	3.41342	0.131383
$676$	0	0
$677$	−1.91135	−0.0734590	−0.0367295	−	0.999325i	$-0.511694\pi$
$677$	−1.91135	−0.0734590	−0.0367295	+	0.999325i	$0.511694\pi$
$678$	0	0
$679$	−10.4959	−0.402794
$680$	0	0
$681$	−1.09869	−0.0421021
$682$	0	0
$683$	−40.2721	−1.54097	−0.770485	−	0.637458i	$-0.779986\pi$
$683$	−40.2721	−1.54097	−0.770485	+	0.637458i	$0.779986\pi$
$684$	0	0
$685$	30.0388	1.14772
$686$	0	0
$687$	10.2950	0.392780
$688$	0	0
$689$	7.49955	0.285710
$690$	0	0
$691$	−16.4765	−0.626795	−0.313397	−	0.949622i	$-0.601467\pi$
$691$	−16.4765	−0.626795	−0.313397	+	0.949622i	$0.601467\pi$
$692$	0	0
$693$	−2.90552	−0.110372
$694$	0	0
$695$	29.3726	1.11416
$696$	0	0
$697$	6.07163	0.229979
$698$	0	0
$699$	2.20850	0.0835331
$700$	0	0
$701$	20.9375	0.790797	0.395398	−	0.918510i	$-0.370607\pi$
$701$	20.9375	0.790797	0.395398	+	0.918510i	$0.370607\pi$
$702$	0	0
$703$	−9.36695	−0.353281
$704$	0	0
$705$	3.60942	0.135939
$706$	0	0
$707$	−11.3424	−0.426576
$708$	0	0
$709$	−38.1386	−1.43233	−0.716163	−	0.697934i	$-0.754103\pi$
$709$	−38.1386	−1.43233	−0.716163	+	0.697934i	$0.754103\pi$
$710$	0	0
$711$	2.63398	0.0987821
$712$	0	0
$713$	12.5505	0.470021
$714$	0	0
$715$	4.02784	0.150633
$716$	0	0
$717$	−11.7362	−0.438296
$718$	0	0
$719$	−18.0836	−0.674405	−0.337202	−	0.941432i	$-0.609481\pi$
$719$	−18.0836	−0.674405	−0.337202	+	0.941432i	$0.609481\pi$
$720$	0	0
$721$	32.4711	1.20929
$722$	0	0
$723$	5.52557	0.205498
$724$	0	0
$725$	1.30259	0.0483769
$726$	0	0
$727$	38.3635	1.42282	0.711412	−	0.702775i	$-0.248056\pi$
$727$	38.3635	1.42282	0.711412	+	0.702775i	$0.248056\pi$
$728$	0	0
$729$	−9.19556	−0.340576
$730$	0	0
$731$	61.8812	2.28876
$732$	0	0
$733$	46.2166	1.70705	0.853524	−	0.521054i	$-0.174461\pi$
$733$	46.2166	1.70705	0.853524	+	0.521054i	$0.174461\pi$
$734$	0	0
$735$	2.15899	0.0796355
$736$	0	0
$737$	4.34201	0.159940
$738$	0	0
$739$	22.0464	0.810991	0.405496	−	0.914097i	$-0.367099\pi$
$739$	22.0464	0.810991	0.405496	+	0.914097i	$0.367099\pi$
$740$	0	0
$741$	−2.12309	−0.0779936
$742$	0	0
$743$	17.7743	0.652077	0.326039	−	0.945356i	$-0.394286\pi$
$743$	17.7743	0.652077	0.326039	+	0.945356i	$0.394286\pi$
$744$	0	0
$745$	30.5987	1.12105
$746$	0	0
$747$	5.28127	0.193231
$748$	0	0
$749$	−34.5875	−1.26380
$750$	0	0
$751$	40.5750	1.48060	0.740301	−	0.672275i	$-0.234683\pi$
$751$	40.5750	1.48060	0.740301	+	0.672275i	$0.234683\pi$
$752$	0	0
$753$	−0.659108	−0.0240192
$754$	0	0
$755$	−8.69689	−0.316512
$756$	0	0
$757$	46.5174	1.69070	0.845352	−	0.534210i	$-0.179391\pi$
$757$	46.5174	1.69070	0.845352	+	0.534210i	$0.179391\pi$
$758$	0	0
$759$	2.16095	0.0784376
$760$	0	0
$761$	−36.6586	−1.32887	−0.664436	−	0.747345i	$-0.731328\pi$
$761$	−36.6586	−1.32887	−0.664436	+	0.747345i	$0.731328\pi$
$762$	0	0
$763$	−13.0508	−0.472471
$764$	0	0
$765$	39.8817	1.44193
$766$	0	0
$767$	−52.1100	−1.88158
$768$	0	0
$769$	13.5212	0.487588	0.243794	−	0.969827i	$-0.421608\pi$
$769$	13.5212	0.487588	0.243794	+	0.969827i	$0.421608\pi$
$770$	0	0
$771$	−7.40120	−0.266548
$772$	0	0
$773$	27.5500	0.990906	0.495453	−	0.868635i	$-0.335002\pi$
$773$	27.5500	0.990906	0.495453	+	0.868635i	$0.335002\pi$
$774$	0	0
$775$	−1.64861	−0.0592198
$776$	0	0
$777$	13.3424	0.478655
$778$	0	0
$779$	−0.982363	−0.0351968
$780$	0	0
$781$	−1.22168	−0.0437151
$782$	0	0
$783$	4.43354	0.158442
$784$	0	0
$785$	5.52115	0.197058
$786$	0	0
$787$	−37.7891	−1.34704	−0.673518	−	0.739171i	$-0.735218\pi$
$787$	−37.7891	−1.34704	−0.673518	+	0.739171i	$0.735218\pi$
$788$	0	0
$789$	−11.9143	−0.424161
$790$	0	0
$791$	−19.1394	−0.680518
$792$	0	0
$793$	−21.7284	−0.771598
$794$	0	0
$795$	−3.16735	−0.112334
$796$	0	0
$797$	3.43431	0.121649	0.0608247	−	0.998148i	$-0.480627\pi$
$797$	3.43431	0.121649	0.0608247	+	0.998148i	$0.480627\pi$
$798$	0	0
$799$	15.0519	0.532499
$800$	0	0
$801$	−9.91735	−0.350412
$802$	0	0
$803$	1.97476	0.0696877
$804$	0	0
$805$	−43.9723	−1.54982
$806$	0	0
$807$	−1.05383	−0.0370967
$808$	0	0
$809$	14.2673	0.501612	0.250806	−	0.968037i	$-0.419304\pi$
$809$	14.2673	0.501612	0.250806	+	0.968037i	$0.419304\pi$
$810$	0	0
$811$	6.73142	0.236372	0.118186	−	0.992991i	$-0.462292\pi$
$811$	6.73142	0.236372	0.118186	+	0.992991i	$0.462292\pi$
$812$	0	0
$813$	−15.1338	−0.530766
$814$	0	0
$815$	42.0285	1.47219
$816$	0	0
$817$	−10.0121	−0.350279
$818$	0	0
$819$	−21.7628	−0.760453
$820$	0	0
$821$	−2.86039	−0.0998282	−0.0499141	−	0.998754i	$-0.515895\pi$
$821$	−2.86039	−0.0998282	−0.0499141	+	0.998754i	$0.515895\pi$
$822$	0	0
$823$	−9.96452	−0.347341	−0.173671	−	0.984804i	$-0.555563\pi$
$823$	−9.96452	−0.347341	−0.173671	+	0.984804i	$0.555563\pi$
$824$	0	0
$825$	−0.283858	−0.00988266
$826$	0	0
$827$	−5.15974	−0.179422	−0.0897109	−	0.995968i	$-0.528594\pi$
$827$	−5.15974	−0.179422	−0.0897109	+	0.995968i	$0.528594\pi$
$828$	0	0
$829$	19.1798	0.666141	0.333070	−	0.942902i	$-0.391915\pi$
$829$	19.1798	0.666141	0.333070	+	0.942902i	$0.391915\pi$
$830$	0	0
$831$	3.24202	0.112464
$832$	0	0
$833$	9.00337	0.311948
$834$	0	0
$835$	11.9466	0.413428
$836$	0	0
$837$	−5.61126	−0.193954
$838$	0	0
$839$	−10.3056	−0.355790	−0.177895	−	0.984049i	$-0.556929\pi$
$839$	−10.3056	−0.355790	−0.177895	+	0.984049i	$0.556929\pi$
$840$	0	0
$841$	−27.3081	−0.941660
$842$	0	0
$843$	5.05423	0.174077
$844$	0	0
$845$	−1.67808	−0.0577276
$846$	0	0
$847$	−25.3818	−0.872129
$848$	0	0
$849$	4.62871	0.158857
$850$	0	0
$851$	71.4112	2.44794
$852$	0	0
$853$	0.0629119	0.00215406	0.00107703	−	0.999999i	$-0.499657\pi$
$853$	0.0629119	0.00215406	0.00107703	+	0.999999i	$0.499657\pi$
$854$	0	0
$855$	−6.45269	−0.220677
$856$	0	0
$857$	12.4923	0.426730	0.213365	−	0.976973i	$-0.431558\pi$
$857$	12.4923	0.426730	0.213365	+	0.976973i	$0.431558\pi$
$858$	0	0
$859$	23.2530	0.793383	0.396692	−	0.917952i	$-0.370158\pi$
$859$	23.2530	0.793383	0.396692	+	0.917952i	$0.370158\pi$
$860$	0	0
$861$	1.39929	0.0476875
$862$	0	0
$863$	18.0492	0.614402	0.307201	−	0.951645i	$-0.400608\pi$
$863$	18.0492	0.614402	0.307201	+	0.951645i	$0.400608\pi$
$864$	0	0
$865$	1.28428	0.0436669
$866$	0	0
$867$	−12.8260	−0.435595
$868$	0	0
$869$	−0.468519	−0.0158934
$870$	0	0
$871$	32.5223	1.10198
$872$	0	0
$873$	11.7421	0.397411
$874$	0	0
$875$	−23.0630	−0.779671
$876$	0	0
$877$	−34.3261	−1.15911	−0.579555	−	0.814933i	$-0.696774\pi$
$877$	−34.3261	−1.15911	−0.579555	+	0.814933i	$0.696774\pi$
$878$	0	0
$879$	0.0285456	0.000962818 0
$880$	0	0
$881$	28.3318	0.954522	0.477261	−	0.878762i	$-0.341630\pi$
$881$	28.3318	0.954522	0.477261	+	0.878762i	$0.341630\pi$
$882$	0	0
$883$	−13.8750	−0.466931	−0.233466	−	0.972365i	$-0.575007\pi$
$883$	−13.8750	−0.466931	−0.233466	+	0.972365i	$0.575007\pi$
$884$	0	0
$885$	22.0081	0.739793
$886$	0	0
$887$	14.7425	0.495005	0.247502	−	0.968887i	$-0.420390\pi$
$887$	14.7425	0.495005	0.247502	+	0.968887i	$0.420390\pi$
$888$	0	0
$889$	6.23014	0.208952
$890$	0	0
$891$	2.73606	0.0916615
$892$	0	0
$893$	−2.43534	−0.0814955
$894$	0	0
$895$	15.1707	0.507102
$896$	0	0
$897$	16.1859	0.540430
$898$	0	0
$899$	−2.14130	−0.0714165
$900$	0	0
$901$	−13.2084	−0.440037
$902$	0	0
$903$	14.2613	0.474587
$904$	0	0
$905$	9.75082	0.324128
$906$	0	0
$907$	3.40751	0.113144	0.0565722	−	0.998399i	$-0.481983\pi$
$907$	3.40751	0.113144	0.0565722	+	0.998399i	$0.481983\pi$
$908$	0	0
$909$	12.6892	0.420875
$910$	0	0
$911$	−16.4500	−0.545014	−0.272507	−	0.962154i	$-0.587853\pi$
$911$	−16.4500	−0.545014	−0.272507	+	0.962154i	$0.587853\pi$
$912$	0	0
$913$	−0.939403	−0.0310897
$914$	0	0
$915$	9.17674	0.303374
$916$	0	0
$917$	−6.78567	−0.224083
$918$	0	0
$919$	−6.53585	−0.215598	−0.107799	−	0.994173i	$-0.534380\pi$
$919$	−6.53585	−0.215598	−0.107799	+	0.994173i	$0.534380\pi$
$920$	0	0
$921$	15.0004	0.494280
$922$	0	0
$923$	−9.15057	−0.301195
$924$	0	0
$925$	−9.38041	−0.308426
$926$	0	0
$927$	−36.3267	−1.19313
$928$	0	0
$929$	37.2527	1.22222	0.611111	−	0.791545i	$-0.290723\pi$
$929$	37.2527	1.22222	0.611111	+	0.791545i	$0.290723\pi$
$930$	0	0
$931$	−1.45671	−0.0477416
$932$	0	0
$933$	12.9565	0.424177
$934$	0	0
$935$	−7.09394	−0.231997
$936$	0	0
$937$	20.7557	0.678061	0.339030	−	0.940775i	$-0.389901\pi$
$937$	20.7557	0.678061	0.339030	+	0.940775i	$0.389901\pi$
$938$	0	0
$939$	9.78609	0.319357
$940$	0	0
$941$	−44.0633	−1.43642	−0.718211	−	0.695825i	$-0.755039\pi$
$941$	−44.0633	−1.43642	−0.718211	+	0.695825i	$0.755039\pi$
$942$	0	0
$943$	7.48927	0.243884
$944$	0	0
$945$	19.6597	0.639532
$946$	0	0
$947$	−13.0438	−0.423867	−0.211933	−	0.977284i	$-0.567976\pi$
$947$	−13.0438	−0.423867	−0.211933	+	0.977284i	$0.567976\pi$
$948$	0	0
$949$	14.7912	0.480144
$950$	0	0
$951$	12.2703	0.397892
$952$	0	0
$953$	9.95984	0.322631	0.161315	−	0.986903i	$-0.448426\pi$
$953$	9.95984	0.322631	0.161315	+	0.986903i	$0.448426\pi$
$954$	0	0
$955$	−25.7067	−0.831848
$956$	0	0
$957$	−0.368690	−0.0119181
$958$	0	0
$959$	28.8695	0.932245
$960$	0	0
$961$	−28.2899	−0.912577
$962$	0	0
$963$	38.6944	1.24691
$964$	0	0
$965$	26.6181	0.856866
$966$	0	0
$967$	41.6459	1.33924	0.669620	−	0.742704i	$-0.266457\pi$
$967$	41.6459	1.33924	0.669620	+	0.742704i	$0.266457\pi$
$968$	0	0
$969$	3.73925	0.120122
$970$	0	0
$971$	−25.5000	−0.818335	−0.409167	−	0.912459i	$-0.634181\pi$
$971$	−25.5000	−0.818335	−0.409167	+	0.912459i	$0.634181\pi$
$972$	0	0
$973$	28.2292	0.904985
$974$	0	0
$975$	−2.12614	−0.0680909
$976$	0	0
$977$	−56.5702	−1.80984	−0.904920	−	0.425581i	$-0.860070\pi$
$977$	−56.5702	−1.80984	−0.904920	+	0.425581i	$0.860070\pi$
$978$	0	0
$979$	1.76404	0.0563791
$980$	0	0
$981$	14.6005	0.466157
$982$	0	0
$983$	2.46370	0.0785798	0.0392899	−	0.999228i	$-0.487490\pi$
$983$	2.46370	0.0785798	0.0392899	+	0.999228i	$0.487490\pi$
$984$	0	0
$985$	−26.0875	−0.831216
$986$	0	0
$987$	3.46892	0.110417
$988$	0	0
$989$	76.3296	2.42714
$990$	0	0
$991$	10.4126	0.330767	0.165383	−	0.986229i	$-0.447114\pi$
$991$	10.4126	0.330767	0.165383	+	0.986229i	$0.447114\pi$
$992$	0	0
$993$	−6.04807	−0.191930
$994$	0	0
$995$	54.2349	1.71936
$996$	0	0
$997$	34.3528	1.08797	0.543983	−	0.839096i	$-0.316916\pi$
$997$	34.3528	1.08797	0.543983	+	0.839096i	$0.316916\pi$
$998$	0	0
$999$	−31.9275	−1.01014

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.g.1.12	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.g.1.12	✓	27	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q-0.604994 q^{3} +2.44978 q^{5} +2.35442 q^{7} -2.63398 q^{9} +O(q^{10})\)	\(q-0.604994 q^{3} +2.44978 q^{5} +2.35442 q^{7} -2.63398 q^{9} +0.468519 q^{11} +3.50927 q^{13} -1.48210 q^{15} -6.18064 q^{17} +1.00000 q^{19} -1.42441 q^{21} -7.62373 q^{23} +1.00144 q^{25} +3.40852 q^{27} +1.30072 q^{29} -1.64624 q^{31} -0.283451 q^{33} +5.76782 q^{35} -9.36695 q^{37} -2.12309 q^{39} -0.982363 q^{41} -10.0121 q^{43} -6.45269 q^{45} -2.43534 q^{47} -1.45671 q^{49} +3.73925 q^{51} +2.13707 q^{53} +1.14777 q^{55} -0.604994 q^{57} -14.8492 q^{59} -6.19170 q^{61} -6.20150 q^{63} +8.59696 q^{65} +9.26754 q^{67} +4.61231 q^{69} -2.60754 q^{71} +4.21490 q^{73} -0.605863 q^{75} +1.10309 q^{77} -1.00000 q^{79} +5.83981 q^{81} -2.00505 q^{83} -15.1412 q^{85} -0.786928 q^{87} +3.76515 q^{89} +8.26231 q^{91} +0.995968 q^{93} +2.44978 q^{95} -4.45794 q^{97} -1.23407 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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